Editing Appendix/Ramblings/RiemannB74C692 (section)

==Quantitative Animation==

===rotation_euler_Z===
Whether the observer is viewing the system from an inertial frame of reference or a rotating frame of reference, this ''animation section'' specifies the rate at which the ellipsoid spins and the rate at which the "laboratory" is spinning.  Three sets of data must be specified:  
<ol>
<li>The "<font color="maroon">First</font>_rotation_euler_Z-input-array", which specifies TIME.</li>
<li>The "<font color="maroon">First</font>_rotation_euler_Z-output-array", which specifies how the ANGLE of orientation of the ellipsoid varies with TIME.</li>
<li>The "<font color="maroon">Frame</font>_rotation_euler_Z-output-array", which specifies how the spin ANGLE of the laboratory frame varies with TIME.</li>
</ol>

We have chosen to specify the time-varying angle in increments of 60&deg; and to end the animation sequence after the ellipsoid (or, alternatively, the laboratory frame) has completed exactly <b>five</b> full spins.  As a result &#8212; ignoring any zero-point offsets &#8212; one of the ANGLE arrays will vary from 0&deg; to 5 &times; 360&deg; = 1800&deg; in 60&deg; intervals (if spinning counter-clock-wise) or will vary from 1800&deg; to 0&deg; in -60&deg; intervals (if spinning clock-wise).  Again ignoring any zero-point offset, all the data values in the other ANGLE array will be 0&deg;.  The number of discrete data values that must be specified for each of the three arrays is therefore, by choice, count = (1800/60)+1 = 31.

All we have left to do is specify the TIME interval that is equivalent to a 60&deg; turn of the ellipsoid, as viewed from the inertial frame of reference.  Given that one cycle of the wall-clock's "minute" hand represents a frequency of <math>~[\pi G \rho]^{1 / 2}</math> and, furthermore by choice, we equate one cycle of the wall-clock's "minute" hand to TIME = 4 in COLLADA time units (approximately 4 seconds of real time), we know that <b>one</b> complete spin of the ellipsoid is  equivalent to 4/|&Omega;<sub>EFE</sub>| units of TIME.  Hence the ellipsoid will turn through 60&deg; in a time, &Delta;T<sub>&Omega;</sub> = 4/[6|&Omega;<sub>EFE</sub>|] and it will take TIME = 20/|&Omega;<sub>EFE</sub>| to complete five full spins.

<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="4">''Direct''</th>
</tr>
<tr>
  <td align="center">Model</td>
  <td align="center">&Omega;<sub>EFE</sub></td>
  <td align="center">&Delta;T<sub>&Omega;</sub></td>
  <td align="center">5 Spins</td>
</tr>
<tr>
  <td align="center">b41c385</td>
  <td align="center">0.547874</td>
  <td align="center">1.2168</td>
  <td align="center">36.505</td>
</tr>
<tr>
  <td align="center" bgcolor="lightblue">b74c692</td>
  <td align="center" bgcolor="lightblue">0.638747</td>
  <td align="center" bgcolor="lightblue">1.0437</td>
  <td align="center" bgcolor="lightblue">31.311</td>
</tr>
<tr>
  <td align="center">b90c333</td>
  <td align="center">0.447158</td>
  <td align="center">1.4909</td>
  <td align="center">44.727</td>
</tr>
<tr>
  <td align="center">b28c256</td>
  <td align="center">0.456676</td>
  <td align="center">1.4598</td>
  <td align="center">43.795</td>
</tr>
</table>

===location_X===
Here we illustrate the motion of one Lagrangian fluid element as it moves along an elliptical path (b/a aspect ratio) in the equatorial plane of the Riemann ellipsoid. First we acknowledge that every fluid element completes one full orbit in a COLLADA-based time of, TIME<sub>&lambda;</sub> = 4/|&lambda;<sub>EFE</sub>|. Dividing the orbit into 50 equally spaced intervals of time therefore specifies a time increment, &Delta;T<sub>&lambda;</sub> = 4/(50 |&lambda;<sub>EFE</sub>|).  Drawing from a [[ThreeDimensionalConfigurations/RiemannStype#Feeding_a_3D_Animation|related accompanying discussion]], we recognize that at each of these points in time &#8212; that is, for n = 1 &rarr; 50 &#8212; the x and y coordinate locations of the fluid element are given by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_n</math>
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\cos\biggl[-2\pi (n-1)\frac{\Delta T_\lambda}{4} \cdot \lambda_\mathrm{EFE}\biggr]</math>
  </td>
  <td align="center" rowspan="2">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~y_n</math>
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\biggl(\frac{b}{a}\biggr)\sin\biggl[- 2\pi (n-1)\frac{\Delta T_\lambda}{4}  \cdot \lambda_\mathrm{EFE} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\cos\biggl[-\frac{2\pi (n-1)}{50} \cdot \frac{\lambda_\mathrm{EFE}}{|\lambda_\mathrm{EFE}|} \biggr]</math>
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\biggl(\frac{b}{a}\biggr)\sin\biggl[ -\frac{2\pi (n-1)}{50} \cdot \frac{\lambda_\mathrm{EFE}}{|\lambda_\mathrm{EFE}|} \biggr] </math>
  </td>
</tr>
</table>
where we have assumed that the length of the longest axis of the ellipsoid is, a = 1.  Note that, in the argument of these trigonometric functions, the sign of the frequency, &lambda;<sub>EFE</sub>, determines whether the fluid motion is regrograde (&lambda;<sub>EFE</sub> intrinsically positive) or prograde (&lambda;<sub>EFE</sub> intrinsically negative).

<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="6">''Direct''</th>
</tr>
<tr>
  <td align="center">Model</td>
  <td align="center">&lambda;<sub>EFE</sub></td>
  <td align="center">&Delta;T<sub>&lambda;</sub></td>
  <td align="center">1 Orbit</td>
  <td align="center">Frequency<br />Ratio<br />5 &times; |&lambda;<sub>EFE</sub>/&Omega;<sub>EFE</sub>|</td>
  <td align="center">Count</td>
</tr>
<tr>
  <td align="center">b41c385</td>
  <td align="center">0.079886</td>
  <td align="center">1.0014</td>
  <td align="center">50.071</td>
  <td align="center">0.7291</td>
  <td align="center">37</td>
</tr>
<tr>
  <td align="center" bgcolor="lightblue">b74c692</td>
  <td align="center" bgcolor="lightblue">0.217773</td>
  <td align="center" bgcolor="lightblue">0.3674</td>
  <td align="center" bgcolor="lightblue">18.368</td>
  <td align="center" bgcolor="lightblue">1.7046</td>
  <td align="center" bgcolor="lightblue">86</td>
</tr>
<tr>
  <td align="center">b90c333</td>
  <td align="center">-0.221411</td>
  <td align="center">0.3613</td>
  <td align="center">18.066</td>
  <td align="center">2.4758</td>
  <td align="center">124</td>
</tr>
<tr>
  <td align="center">b28c256</td>
  <td align="center">0.020692</td>
  <td align="center">3.8662</td>
  <td align="center">193.31</td>
  <td align="center">0.2266</td>
  <td align="center">12</td>
</tr>
</table>

Here are some x-y coordinate pairs for a couple of model examples:
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="8">''Direct''</th>
</tr>
<tr>
  <td align="center" rowspan="2">n</td>
  <td align="center" colspan="2">Axisymmetric</td>
  <td align="center" colspan="1">b41c385</td>
  <td align="center" colspan="1" bgcolor="lightblue">b74c692</td>
  <td align="center" colspan="1">b90c333</td>
  <td align="center" colspan="1">b28c256</td>
</tr>
<tr>
  <td align="center">x<sub>0</sub></td>
  <td align="center">y<sub>0</sub></td>
  <td align="center">y = 0.41 &times; y<sub>0</sub></td>
  <td align="center">y = 0.74 &times; y<sub>0</sub></td>
  <td align="center">y = 0.90 &times; y<sub>0</sub></td>
  <td align="center">y = 0.28 &times; y<sub>0</sub></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="right">1.0000</td>
  <td align="right">0.0000</td>
  <td align="right">0.0000</td>
  <td align="right" bgcolor="lightblue">0.0000</td>
  <td align="right">0.0000</td>
  <td align="right">0.0000</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="right">0.9921</td>
  <td align="right">-0.1253</td>
  <td align="right">-0.0514</td>
  <td align="right" bgcolor="lightblue">-0.0927</td>
  <td align="right">-0.1128</td>
  <td align="right">-0.0351</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="right">0.9686</td>
  <td align="right">-0.2487</td>
  <td align="right">-0.1020</td>
  <td align="right" bgcolor="lightblue">-0.1840</td>
  <td align="right">-0.2238</td>
  <td align="right">-0.0696</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="right">0.9298</td>
  <td align="right">-0.3681</td>
  <td align="right">-0.1509</td>
  <td align="right" bgcolor="lightblue">-0.2724</td>
  <td align="right">-0.3313</td>
  <td align="right">-0.1031</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="right">0.8763</td>
  <td align="right">-0.4818</td>
  <td align="right">-0.1975</td>
  <td align="right" bgcolor="lightblue">-0.3565</td>
  <td align="right">-0.4336</td>
  <td align="right">-0.1349</td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="right">0.8090</td>
  <td align="right">-0.5878</td>
  <td align="right">-0.2410</td>
  <td align="right" bgcolor="lightblue">-0.4350</td>
  <td align="right">-0.5290</td>
  <td align="right">-0.1646</td>
</tr>
<tr>
  <td align="center">7</td>
  <td align="right">0.7290</td>
  <td align="right">-0.6845</td>
  <td align="right">-0.2807</td>
  <td align="right" bgcolor="lightblue">-0.5066</td>
  <td align="right">-0.6161</td>
  <td align="right">-0.1917</td>
</tr>
<tr>
  <td align="center">8</td>
  <td align="right">0.6374</td>
  <td align="right">-0.7705</td>
  <td align="right">-0.3159</td>
  <td align="right" bgcolor="lightblue">-0.5702</td>
  <td align="right">-0.6935</td>
  <td align="right">-0.2157</td>
</tr>
<tr>
  <td align="center">9</td>
  <td align="right">0.5358</td>
  <td align="right">-0.8443</td>
  <td align="right">-0.3462</td>
  <td align="right" bgcolor="lightblue">-0.6248</td>
  <td align="right">-0.7599</td>
  <td align="right">-0.2364</td>
</tr>
<tr>
  <td align="center">10</td>
  <td align="right">0.4258</td>
  <td align="right">-0.9048</td>
  <td align="right">-0.3710</td>
  <td align="right" bgcolor="lightblue">-0.6696</td>
  <td align="right">-0.8143</td>
  <td align="right">-0.2533</td>
</tr>
<tr>
  <td align="center">11</td>
  <td align="right">0.3090</td>
  <td align="right">-0.9511</td>
  <td align="right">-0.3899</td>
  <td align="right" bgcolor="lightblue">-0.7038</td>
  <td align="right">-0.8560</td>
  <td align="right">-0.2663</td>
</tr>
<tr>
  <td align="center">12</td>
  <td align="right">0.1874</td>
  <td align="right">-0.9823</td>
  <td align="right">-0.4027</td>
  <td align="right" bgcolor="lightblue">-0.7269</td>
  <td align="right">-0.8841</td>
  <td align="right">-0.2750</td>
</tr>
<tr>
  <td align="center">13</td>
  <td align="right">0.0628</td>
  <td align="right">-0.9980</td>
  <td align="right">-0.4092</td>
  <td align="right" bgcolor="lightblue">-0.7385</td>
  <td align="right">-0.8982</td>
  <td align="right">-0.2794</td>
</tr>
</table>

How many complete (and fractional) orbits does each Lagrangian fluid element complete in the time it takes the ellipsoid to complete five spin periods?  And, as a result, how many discrete steps in time does it traverse, assuming that each orbit has been dissected into 50 pieces?  The answer to the first question is given by dividing "5 Spins" by "1 Orbit"; in the table above, we label this quantity as, "Frequency Ratio," given that it is equivalent to |5 &times; &lambda;<sub>EFE</sub>/&Omega;<sub>EFE</sub>|.  And the answer to the second question is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
Count
  </td>
  <td align="center">
=
  </td>
  <td align="left">
(50 &times; Frequency_Ratio) + 1 .
  </td>
</tr>
</table>

===Cube_quaternion===

Here we construct the wall-clock.  First we recognize that the time required for the ellipsoid to complete "5 spins" is, as calculated above, TIME = 20/|&Omega;<sub>EFE</sub>|.  We have also dictated that the "minute hand" on the clock will complete one full cycle &#8212; that is, it will spin through an angle that starts at 0&deg; and runs to &gamma; = 360&deg; &#8212; every 4 TIME units.  While watching the ellipsoid spin five times, the "minute hand" must complete M = 5/|&Omega;<sub>EFE</sub>| cycles.

We have chosen to model the motion of the "minute hand" by breaking each cycle (360&deg;) into 16 equal divisions, that is, we have chosen to set &Delta;&gamma; = 360&deg;/16 = 22.5&deg;.  This also means that the number of discrete time values will be INT(M &times; 16) + 1 = INT(80/|&Omega;<sub>EFE</sub>|) + 1.

<table border="0" align="center" cellpadding="10"><tr>
<td align="left">

<table border="1" align="center" cellpadding="8">
<tr>
 <td align="center" colspan="3">To be used in the<br />Quaternion Array</td>
</tr>
<tr>
  <td align="center">&gamma; (degrees)</td>
  <td align="center">sin &gamma;</td>
  <td align="center">cos &gamma;</td>
</tr>
<tr>
  <td align="right">-90.0</td>
  <td align="right">-1.0000</td>
  <td align="right">0.0000</td>
</tr>
<tr>
  <td align="right">-112.5</td>
  <td align="right">-0.9239</td>
  <td align="right">-0.3827</td>
</tr>
<tr>
  <td align="right">-135.0</td>
  <td align="right">-0.7071</td>
  <td align="right">-0.7071</td>
</tr>
<tr>
  <td align="right">-157.5</td>
  <td align="right">-0.3827</td>
  <td align="right">-0.9239</td>
</tr>
<tr>
  <td align="right">-180.0</td>
  <td align="right">0.0000</td>
  <td align="right">-1.0000</td>
</tr>
<tr>
  <td align="right">-202.5</td>
  <td align="right">0.3827</td>
  <td align="right">-0.9239</td>
</tr>
<tr>
  <td align="right">-225.0</td>
  <td align="right">0.7071</td>
  <td align="right">-0.7071</td>
</tr>
<tr>
  <td align="right">-247.5</td>
  <td align="right">0.9239</td>
  <td align="right">-0.3827</td>
</tr>
<tr>
  <td align="right">-270.0</td>
  <td align="right">1.0000</td>
  <td align="right">0.0000</td>
</tr>
<tr>
  <td align="right">-292.5</td>
  <td align="right">0.9239</td>
  <td align="right">0.3827</td>
</tr>
<tr>
  <td align="right">-315.0</td>
  <td align="right">0.7071</td>
  <td align="right">0.7071</td>
</tr>
<tr>
  <td align="right">-337.5</td>
  <td align="right">0.3827</td>
  <td align="right">0.9239</td>
</tr>
<tr>
  <td align="right">0.0</td>
  <td align="right">0.0000</td>
  <td align="right">1.0000</td>
</tr>
<tr>
  <td align="right">-22.5</td>
  <td align="right">-0.3827</td>
  <td align="right">0.9239</td>
</tr>
<tr>
  <td align="right">-45.0</td>
  <td align="right">-0.7071</td>
  <td align="right">0.7071</td>
</tr>
<tr>
  <td align="right">-67.5</td>
  <td align="right">-0.9239</td>
  <td align="right">0.3827</td>
</tr>
</table>

</td>
<td align="left">

<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="4">''Clock''</th>
</tr>
<tr>
  <td align="center">Model</td>
  <td align="center">"Minute Hand"<br />Cycles</td>
  <td align="center">Discrete<br />Steps</td>
  <td align="center">N/A</td>
</tr>
<tr>
  <td align="center">b41c385</td>
  <td align="center">9.1261</td>
  <td align="center">147</td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center" bgcolor="lightblue">b74c692</td>
  <td align="center" bgcolor="lightblue">7.8278</td>
  <td align="center" bgcolor="lightblue">126</td>
  <td align="center" bgcolor="lightblue">---</td>
</tr>
<tr>
  <td align="center">b90c333</td>
  <td align="center">11.1817</td>
  <td align="center">179</td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center">b28c256</td>
  <td align="center">10.9487</td>
  <td align="center">176</td>
  <td align="center"></td>
</tr>
</table>
</td></tr></table>

As we have [[Appendix/Ramblings/RiemannMeetsOculus#Final_Touches|detailed in a parallel discussion]], at each discrete time step, the equivalent '''<matrix>''' instruction should be of the form, <table border="0" align="center" cellpadding="8"><tr><td align="center"><matrix>[ 0 ] &nbsp; [ 0 ] &nbsp; [ -S<sub>x</sub> ]  &nbsp; [ T<sub>x</sub> ] &nbsp; &nbsp; [ S<sub>y</sub> &middot; sin(&gamma;) ] &nbsp; [ S<sub>y</sub> &middot; cos(&gamma;) ] &nbsp; [ 0 ] &nbsp; [ T<sub>y</sub> ] &nbsp;  &nbsp; [ S<sub>z</sub> &middot; cos(&gamma;) ] &nbsp; [ -S<sub>z</sub> &middot; sin(&gamma;) ] &nbsp; [ 0] &nbsp; [ T<sub>z</sub> ] &nbsp; &nbsp; [ 0 ] &nbsp; [ 0 ] &nbsp; [ 0 ] &nbsp; [ 1 ]</matrix> .</td></tr></table>

Note that after the clock was originally built as a separate object for these visual scenes, I realized that I needed the starting angle for both hands to be &gamma; = +270&deg; = -90&deg; in order for them to be properly registered at 0<sup>h</sup>0<sup>m</sup> at the start of each animation.  This is why the table of signs and cosines (immediately above) starts at -90&deg;.